Chapter 8 Flashcards
What’s the purpose of density matrices?
- Explain entanglement
- Unify classical and quantum info
- Explain decoherence
- Represent uncertainty in quantum states (mixed states)
- Describe subsystems
What properties does a density matrix have?
- Trace of one
- Positive semidefinite (eigenvalues)
What do the diagonal and off-diagonal elements of a density matrix represent?
Diagonal: probability for classical state
Off-diagonal: coherences
What do the eigenvalues of density matrices represent?
Uncertainty or randomness
What’s a pure state?
A state for which an outer product with itself forms a valid density matrix.
Pure state density matrices have a single eigenvalue equal to 1 for which the eigenvector is the pure state.
What’s a convex combination?
Linear combinations where all the coefficients of each density matrix in the sum are positive and they sum to one.
What is the completey mixed state?
1/2(I)
Lies in the centre of Bloch ball
What does the Von Neumann entropy measure?
Mixedness/disorder of state
S(rho) = -T[rho log_2 rho]
What’s the difference between correlated and entangled density matrix?
Correlated: classical correlations, separable (can be written as convex combinations of product states over the two systems)
Entangled: beyond-classical correlations, non-independent, non-separable
How can you write the reduced state of A as a function of the composite state (A+B)?
rho_A= T_B[rho]
How do you take the partial trace?
T_B[AxB]=A T[B]
What does the Von Neumann entropy of a reduced state density matrix represent?
The number of states in B which are correlated with the measurements of the subsystem A.
What is entanglement entropy?
The Von Neumann entropy of a reduced state density matrix for which the full system is a pure state.
Used to quantify the degree of entanglement
between the two subsystems.
What are channels?
Linear operators which transform
density matrices to other valid density matrices even if applied to subsystems.
What conditions must channels obey?
- Trace preserving (TP)
- Completely positive (CP) mapping
Refered to as CPTP linear maps
How is a unitary channel applied?
By conjugating the density matrix with the unitary operator
What is the identity channel?
Conjugating the identity with a unitary operator.
What does the reset channel do?
Resets the qubit to the 0 state.
Represented by a capital Lamda.
What does the dephasing channel do?
Zeros out the off-diagonal density matrix elements leaving the diagonal elements.
Represented by a capital Delta.
What does the depolarizing channel do?
Returns the completely mixed state.
What is the Pauli channel?
Linear combination of Pauli operators each conjuation the density matrix.
